/* SPDX-License-Identifier: SunMicrosystems */
/* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. */

/**
 *
 * This family of functions is a set of functions used by the gamma function
 * procedures as internal functions. Only ``__lgamma`` is accessible via
 * ``gammad.h`` (``gammaf.h``), but should not be accessed directly by a user.
 *
 * Synopsis
 * ========
 *
 * .. code-block:: c
 *
 *     #include <math.h>
 *     float __sin_pif(float x)
 *     double __sin_pi(double x)
 *     float __lgammaf(float x, int *signgamp)
 *     double __lgamma(double x, int *signgamp)
 *
 * Description
 * ===========
 *
 * ``__lgamma`` computes the natural logarithm of the gamma function of
 * :math:`x` and places the sign of the gamma function in the out-pointer
 * :math:`signgamp`.
 *
 * ``__sin_pi`` computes the sine of the input value after it was multiplied by
 * :math:`\pi`. The procedure is only called in the range
 * :math:`[-2^{52},-2^{-70}]`, calling it outside of this range may result in
 * unexpected results.
 *
 * Mathematical Function
 * =====================
 *
 * .. math::
 *
 *    \_\_sin\_pi(x) &\approx sin(\pi \cdot x)  \\
 *    \_\_lgamma(x) &\approx \ln{|\Gamma(x)|} = \ln{\left|\int_{0}^{\infty}e^{-t}t^{x-1}dt\right|}  \\
 *    \_\_lgamma\_signgamp(x) &= \left\{\begin{array}{ll} +1, & +0 \leq  \Gamma(x) \\ -1, & -0 \geq \Gamma(x) \end{array}\right.
 *
 * Returns
 * =======
 *
 * ``__lgamma`` returns the natural logarithm of the gamma function of
 * :math:`x` and places the sign of the gamma function in the out-pointer
 * :math:`signgamp`.
 *
 * ``__sin_pi`` returns the sine of :math:`\pi \cdot x`.
 *
 * Exceptions
 * ==========
 *
 * Do not raise useful exceptions.
 *
 * .. May raise ``underflow`` exception.
 *
 * Output map
 * ==========
 *
 * The output maps are in the respective external functions :ref:`lgamma` and
 * :ref:`tgamma`.
 *
 *///

/* __lgamma_r(x, signgamp)
 * Reentrant version of the logarithm of the Gamma function
 * with user provide pointer for the sign of Gamma(x).
 *
 * Method:
 *   1. Argument Reduction for 0 < x <= 8
 *     Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
 *     reduce x to a number in [1.5,2.5] by
 *         lgamma(1+s) = log(s) + lgamma(s)
 *    for example,
 *        lgamma(7.3) = log(6.3) + lgamma(6.3)
 *                = log(6.3*5.3) + lgamma(5.3)
 *                = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
 *   2. Polynomial approximation of lgamma around its
 *    minimun ymin=1.461632144968362245 to maintain monotonicity.
 *    On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
 *        Let z = x-ymin;
 *        lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
 *    where
 *        poly(z) is a 14 degree polynomial.
 *   2. Rational approximation in the primary interval [2,3]
 *    We use the following approximation:
 *        s = x-2.0;
 *        lgamma(x) = 0.5*s + s*P(s)/Q(s)
 *    with accuracy
 *        |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
 *    Our algorithms are based on the following observation
 *
 *                             zeta(2)-1    2    zeta(3)-1    3
 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
 *                                 2                 3
 *
 *    where Euler = 0.5771... is the Euler constant, which is very
 *    close to 0.5.
 *
 *   3. For x>=8, we have
 *    lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
 *    (better formula:
 *       lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
 *    Let z = 1/x, then we approximation
 *        f(z) = lgamma(x) - (x-0.5)(log(x)-1)
 *    by
 *                      3       5             11
 *        w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
 *    where
 *        |w - f(z)| < 2**-58.74
 *
 *   4. For negative x, since (G is gamma function)
 *        -x*G(-x)*G(x) = pi/sin(pi*x),
 *     we have
 *         G(x) = pi/(sin(pi*x)*(-x)*G(-x))
 *    since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
 *    Hence, for x<0, signgam = sign(sin(pi*x)) and
 *        lgamma(x) = log(|Gamma(x)|)
 *              = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
 *    Note: one should avoid compute pi*(-x) directly in the
 *          computation of sin(pi*(-x)).
 *
 *   5. Special Cases
 *        lgamma(2+s) ~ s*(1-Euler) for tiny s
 *        lgamma(1)=lgamma(2)=0
 *        lgamma(x) ~ -log(x) for tiny x
 *        lgamma(0) = lgamma(inf) = inf
 *         lgamma(-integer) = +-inf
 *
 */

#include <math.h>
#include "../../common/tools.h"
#include "gammad.h"
#include "trigd.h"

#ifndef __LIBMCS_DOUBLE_IS_32BITS

static const double
two52 =  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
half  =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
pi    =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
a0    =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
a1    =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
a2    =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
a3    =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
a4    =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
a5    =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
a6    =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
a7    =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
a8    =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
a9    =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
a10   =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
a11   =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
tc    =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
tf    = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
/* tt = -(tail of tf) */
tt    = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
t0    =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
t1    = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
t2    =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
t3    = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
t4    =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
t5    = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
t6    =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
t7    = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
t8    =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
t9    = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
t10   =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
t11   = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
t12   =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
t13   = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
t14   =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
u0    = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
u1    =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
u2    =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
u3    =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
u4    =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
u5    =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
v1    =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
v2    =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
v3    =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
v4    =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
v5    =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
s0    = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
s1    =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
s2    =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
s3    =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
s4    =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
s5    =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
s6    =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
r1    =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
r2    =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
r3    =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
r4    =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
r5    =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
r6    =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
w0    =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
w1    =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
w2    = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
w3    =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
w4    = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
w5    =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
w6    = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */

static const double zero =  0.00000000000000000000e+00;

static double __sin_pi(double x)
{
    double y, z;
    int32_t n, ix;

    GET_HIGH_WORD(ix, x);
    ix &= 0x7fffffff;

    if (ix < 0x3fd00000) {
        return __sin(pi * x, zero, 0);
    }

    y = -x;                         /* x is assume negative */

    /*
     * argument reduction, make sure inexact flag not raised if input
     * is an integer
     */
    z = floor(y);

    if (z != y) {                   /* inexact anyway */
        y  *= 0.5;
        y   = 2.0 * (y - floor(y)); /* y = |x| mod 2.0 */
        n   = (int32_t)(y * 4.0);
    } else {
        z = y + two52;              /* exact */

        GET_LOW_WORD(n, z);
        n &= 1;
        y  = n;
        n <<= 2;
    }

    switch (n) {
    case 0:
        y =  __sin(pi * y, zero, 0);
        break;

    case 1: /* FALLTHRU */
    case 2:
        y =  __cos(pi * (0.5 - y), zero);
        break;

    case 3: /* FALLTHRU */
    case 4:
        y =  __sin(pi * (one - y), zero, 0);
        break;

    case 5: /* FALLTHRU */
    case 6:
        y = -__cos(pi * (y - 1.5), zero);
        break;

    default:
        y =  __sin(pi * (y - 2.0), zero, 0);
        break;
    }

    return -y;
}

double __lgamma(double x, int *signgamp)
{
    double t, y, z, nadj = 0.0, p, p1, p2, p3, q, r, w;
    int32_t i, hx, lx, ix;

    EXTRACT_WORDS(hx, lx, x);

    /* purge off +-inf, NaN, +-0, and negative arguments */
    *signgamp = 1;
    ix = hx & 0x7fffffff;

    if (ix >= 0x7ff00000) {
        return x * x;
    }

    if ((ix | lx) == 0) {
        if(hx < 0) {
            *signgamp = -1;
        }
        return __raise_div_by_zero(zero);
    }

    if (ix < 0x3b900000) { /* |x|<2**-70, return -log(|x|) */
        if (hx < 0) {
            *signgamp = -1;
            return -log(-x);
        } else {
            return -log(x);
        }
    }

    if (hx < 0) {
        if (ix >= 0x43300000) { /* |x|>=2**52, must be -integer */
            return __raise_div_by_zero(zero);
        }

        t = __sin_pi(x);

        if (t == zero) {    /* -integer */
            return __raise_div_by_zero(zero);
        }

        nadj = log(pi / fabs(t * x));

        if (t < zero) {
            *signgamp = -1;
        }

        x = -x;
    }

    /* purge off 1 and 2 */
    if ((((ix - 0x3ff00000) | lx) == 0) || (((ix - 0x40000000) | lx) == 0)) {
        r = 0;
    }
    /* for x < 2.0 */
    else if (ix < 0x40000000) {
        if (ix <= 0x3feccccc) {  /* lgamma(x) = lgamma(x+1)-log(x) */
            r = -log(x);

            if (ix >= 0x3FE76944) {
                y = one - x;
                i = 0;
            } else if (ix >= 0x3FCDA661) {
                y = x - (tc - one);
                i = 1;
            } else {
                y = x;
                i = 2;
            }
        } else {
            r = zero;

            if (ix >= 0x3FFBB4C3) {
                y = 2.0 - x;    /* [1.7316,2] */
                i = 0;
            } else if (ix >= 0x3FF3B4C4) {
                y = x - tc;    /* [1.23,1.73] */
                i = 1;
            } else {
                y = x - one;
                i = 2;
            }
        }

        switch (i) {
        default:    /* FALLTHRU */
        case 0:
            z = y * y;
            p1 = a0 + z * (a2 + z * (a4 + z * (a6 + z * (a8 + z * a10))));
            p2 = z * (a1 + z * (a3 + z * (a5 + z * (a7 + z * (a9 + z * a11)))));
            p  = y * p1 + p2;
            r  += (p - 0.5 * y);
            break;

        case 1:
            z = y * y;
            w = z * y;
            p1 = t0 + w * (t3 + w * (t6 + w * (t9 + w * t12))); /* parallel comp */
            p2 = t1 + w * (t4 + w * (t7 + w * (t10 + w * t13)));
            p3 = t2 + w * (t5 + w * (t8 + w * (t11 + w * t14)));
            p  = z * p1 - (tt - w * (p2 + y * p3));
            r += (tf + p);
            break;

        case 2:
            p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * u5)))));
            p2 = one + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * v5))));
            r += (-0.5 * y + p1 / p2);
            break;
        }
    } else if (ix < 0x40200000) {        /* x < 8.0 */
        i = (int32_t)x;
        y = x - (double)i;
        p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
        q = one + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6)))));
        r = half * y + p / q;
        z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */

        switch (i) {
        case 7:
            z *= (y + 6.0);  /* FALLTHRU */

        case 6:
            z *= (y + 5.0);  /* FALLTHRU */

        case 5:
            z *= (y + 4.0);  /* FALLTHRU */

        case 4:
            z *= (y + 3.0);  /* FALLTHRU */

        case 3:
            z *= (y + 2.0);  /* FALLTHRU */

        default:
            r += log(z);
            break;
        }

    } else if (ix < 0x43900000) {    /* 8.0 <= x < 2**58 */
        t = log(x);
        z = one / x;
        y = z * z;
        w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6)))));
        r = (x - half) * (t - one) + w;
    } else {    /* 2**58 <= x <= inf */
        r =  x * (log(x) - one);
    }

    if (hx < 0) {
        r = nadj - r;
    }

    return r;
}

#endif /* #ifndef __LIBMCS_DOUBLE_IS_32BITS */
